k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an \(O(n\log k)+(1/\epsilon )^{O(2^dk^{1-1/d}\log k)}\) -time \((1+\epsilon )\) -approximation for Euclidean k-center, where d is the dimension. We show for a closely related problem, k-supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k-center and k-supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a \(2^{O(k\log k)}n^2\) time 3-approximation for NUkC in general metrics, and a \(2^{O((k\log k)/\epsilon )}dn\) time \((1+\epsilon )\) -approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.
